The coefficient of x 301 in ( 1 + x ) 500 + x ( 1 + x ) 499 + x 2 ( 1 + x ) 498 + . . . + x 500 is [2023]

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Solution

Q.
The coefficient of $x^{301}$ in ${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + . . . + x^{500}$ is [2023]

1 ${}^{500}{C_{301}}$

2 ${}^{501}{C_{200}}$

3 ${}^{500}{C_{300}}$

4 ${}^{501}{C_{302}}$

Ans.
(2)

$The coefficient of x^{301} in$

${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + \dots + x^{500}$

$= {}^{500}C_{301} + {}^{499}C_{300} + {}^{498}C_{299} + \dots + {}^{199}C_{0}$

$= {}^{500}C_{199} + {}^{499}C_{199} + {}^{498}C_{199} + \dots + {}^{199}C_{199} = {}^{501}C_{200}$

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